Rings of differential operators as enveloping algebras of Hasse-Schmidt derivations

Let k be a commutative ring and A a commutative k-algebra. In this paper we introduce the notion of enveloping algebra of Hasse-Schmidt derivations of A over k and we prove that, under suitable smoothness hypotheses, the canonical map from the above enveloping algebra to the ring of differential operators D-A/k is an isomorphism. This result generalizes the characteristic 0 case in which the ring D(A/k )appears as the enveloping algebra of the Lie-Rinehart algebra of the usual k-derivations of A provided that A is smooth over k. (C) 2019 Elsevier B.V. All rights reserved.